Towards Arbitrage-free Implied Volatility Surfaces with Diffusion Probabilistic Models

Abstract

Implied volatility surfaces (IVS) are essential for option pricing and risk management. Recently, generative deep learning models, such as the Denoising Diffusion Probabilistic Model (DDPM), have gained popularity for generating IVS. However, these machine-generated surfaces do not necessarily meet arbitrage-free constraints, e.g., calendar spread and butterfly arbitrage.
This thesis addresses this limitation by integrating arbitrage-free constraints into the DDPM generation process. We introduce a classifier specifically designed to detect violations of arbitrage-free conditions. During the DDPM’s iterative generation process, the classifier guides the formation of the IVS at each step, effectively minimizing potential arbitrage opportunities. To evaluate the effectiveness of our approach, we compare the extent of arbitrage violations in IVS generated by the standard DDPM with those produced by our enhanced DDPM-Classifier model. The results demonstrate that the DDPM-Classifier significantly reduces arbitrage opportunities and improves the quality of IVS.